%I #15 Aug 02 2022 12:59:58
%S 8,32,512,8192,33554432,8589934592,137438953472,2305843009213693952,
%T 2658455991569831745807614120560689152,
%U 191561942608236107294793378393788647952342390272950272
%N Ultraperfect numbers: 2^(2p - 1), where p is A000043(n).
%C Sum of n-th even perfect number and n-th even superperfect number.
%C Also, sum of n-th perfect number and n-th superperfect number, if there are no odd perfect and odd superperfect numbers, then the n-th perfect number is the difference between a(n) and the n-th superperfect number (see A135652, A135653, A135654 and A135655).
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%F a(n) = 2^(2*A000043(n) - 1). Also, a(n) = 2^A133033(n), if there are no odd perfect numbers. Also, a(n) = A000396(n) + A019279(n), if there are no odd perfect and odd superperfect numbers. Also, a(n) = A000396(n) + A061652(n), if there are no odd perfect numbers, then we can write: perfect number A000396(n) = a(n) - A061652(n).
%F a(n) = A061652(n)*(A000668(n)+1) = A061652(n)*A072868(n). - _Omar E. Pol_, Apr 13 2008
%e a(5)=33554432 because A000043(5)=13 and 2^(2*13 - 1) = 2^25 = 33554432.
%e Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.
%Y Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.
%Y Cf. A000043, A000668, A072868.
%K nonn
%O 1,1
%A _Omar E. Pol_, Apr 13 2008
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